[Avodah] Chazal are Infallible

[Micha Berger]

On Wed, September 13, 2006 6:00 pm, RCM wrote:
: Of course not. A common response is that they knew that these values were not
: exact, but only approximations. They used these values because in halacha
: these approximations are sufficient. I am not sure if this answer can be used
: for all the later meforshim as well as it may be hard to fit into  some of
: their discussions.

This is an oft visited topic here.

Pi, sqrt(2), the number of days in a month or months in a year are all
irrational numbers. Meaning, there is no way to give an exact number, the
exact value requires an infinite number of digits after the decimal point or a
fraction of infinite digits length to represent. Simply can not be done. It is
therefore meaningless to ask why they didn't use an exact value. Estimation is
the only possibility.

The only question that remains is one of shiur: How close is "close enough"
lehalakhah. As RCM concludes as well. (But, IMHO, for slightly less compelling
reasons. I wanted to point out that the issue isn't merely pragmatic, it's the
only possible outcome.)

I think this also addresses RSK's question about approximations. You can't
deduce anything from their using an estimate over the exact value when the
exact value is irrational. As for his later question:
> I know the Vilna Gaon wrote a math book ...
> but exactly where in the Rambam do we see him discuss equations of pi,
> x-squared, and the Pythagorean theorem?

The Rambam's discussion of qidush hachodash doesn't involve any of these, but
does reflect pretty solid knowledge of spherical trig in his translation of
time to degrees. As the Chazon Ish's commentary shows, somewhere among all
diagrams and proofs. It is also hard to believe that someone who knew
Aristotle as well as the Rambam did skimmed over his discussions of geometry.



On Thu, September 14, 2006 3:45 am, Arie Folger wrote:
: Eh, regarding the Rishonim, let me clearly state the opposite. There are
: elements in the Israeli education scene that want to make Tosafot's proof
: (which is actually from an earlier Jewish source, I believe) standard in
: teaching geometry, because students can more readily grasp it than the usualy
: calculations. That this beautiful and simple proof comes from our masters,
: the Rishonim, is of course an important benefit. The secular Israeli teenager
: will be somewhat exposed to their greatness through a subject matter he
: understands.

Back when I was in YU, R' Dr Eliezer/Leon Ehrenpreis used this Tosafos the
first day of calculus. Their use of concentric rings is a step toward the
concept of limits, which in turn is the basis of Newton's original formulation
of calculus. What's missing is the idea of an infinite number of infintesimal
strings. It is therefore an example of "exhaustion" not "limits", much like
the aforementioned approximation of pi by Archemedes.

: I would restate your speculation, saying that 'Hazal were interested in
: stating things in such a way that a non-scientifically trained person could
: apply halakhah. RMB stated the same thing regarding the height of the sukkah,
: IIRC.

I didn't intend to as I have some problem with this sentiment. Are you saying
that the height of the Sukkah is a derabbanan gezeirah far short of the
requirement deOraisa? I thought the rishonim include it as one of the shiurim
that are halakhah leMosheh miSinai.

I believe the height of the sukkah is determined experientially, not
scientifically. IOW, it's not an issue of trig, but of psychology. If the eye
doesn't tend to look at the sekhakh, then one's experience is not that of
sitting in the shade of the sekhakh.

-mi
http://www.aishdas.org/asp